This paper deals with recursive state estimation for the class of discrete time nonlinear systems whose nonlinearity consists of one or more static nonlinear one-variable functions. This class contains several important subclasses. The special structure is exploited to permit accurate computations without an increase in computational cost. The proposed method is compared with standard Extended Kalman Filter, Unscented Kalman Filter and Gauss-Hermite Kalman Filter in three illustrative examples. The results show that it yields good results with small computational cost.